Ian Clark <[email protected]> wrote: > One idea to come out of all this, which may or may not be original, but is > impossible to contemplate addressing with floating-point numbers, is that > inside a tool which is working with real-world observations, the numbers > chiefly of interest will be small ones, in terms of the amount of storage > they occupy, so that when designing iterative methods there may be some > traction in steering away from large numbers into neighbouring small ones.
Finding how to steer of this is one of the points of the super-accumulator -- it keeps a finite bound on the error bars, which allows for well-bounded speed, while allowing intermediates to include both very small and very large values, with rounding to the final format only when it's time to output. > In opposition to that idea, it's worth remembering that between any two > rational numbers there is an uncountable infinity of irrational ones, Not particularly relevant; only a countable number of irrationals can appear as the limit of computations. And some pairs of rationals have no irrationals between them (for example, 0 and the nearest rational to it, which is of the form 1/x, cannot have any irrationals between them, even though (1/x and 1/(x-1)) has an countable infinity of rationals between _them_). Chaitin proved that all of the accessible irrationals are a countable set. (That is, the ones that can be named, defined, specified, or in any way thought about, even when including the ones which can be shown to be uncomputable such as his own "Omega" constant.) "The Tao that can be named is not the true Tao." > and > any attempt to avoid them might reintroduce the very sort of noise I'm > aiming to eliminate. Don't worry -- you can't compute irrationals by a finite process. So you don't have to avoid them, they'll automatically avoid you. :) > But the overwhelming sensation I have at the moment is one of admiration > plus gratitude for the originators (and improvers) of J, for all that > hidden work where nobody imagined it would matter. I am continually amazed. It reminds me of my first real math professor (after junior college). He taught abstract algebra, and in particular he used a language called "Forth" to explore finite group theory. It was very impressive to be able to get my hands on those very abstract concepts. Even more so because I knew Forth before I'd ever thought of learning abstract algebra, and would have never guessed it would become a tool in a mathematician's bag of tricks. J, on the other hand... I can see why people don't learn it, but it's an amazing force multiplier once you do. > Ian Clark -Wm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
